Question

resolva as seguintes equações exponenciais.​

Answer 1

Nas 4 questões que seguem, teremos a mesma abordagem, vamos reescrever a equação dada como uma igualdade de potencias de mesma base.

a)

$7^{5x-13}~=~49\\\\\\\boxed{7^{5x-13}~=~7^2}~~\Rightarrow~Igualdade~entre~potencias~de~mesma~base\\\\\\Pra~que~a~igualdade~seja~verificada,~os~expoentes~tambem~deverao\\ser~iguais,~logo:\\\\\\7\!\!\!\backslash^{5x-13}~=~7\!\!\!\backslash^2\\\\\\5x-13~=~2\\\\\\5x~=~2+13\\\\\\5x~=~15\\\\\\x~=~\dfrac{15}{5}\\\\\\\boxed{x~=~3}$

b)

$5^{2y-2}~=~25\\\\\\\boxed{5^{2y-2}~=~5^2}~~\Rightarrow~Igualdade~entre~potencias~de~mesma~base\\\\\\Pra~que~a~igualdade~seja~verificada,~os~expoentes~tambem~deverao\\ser~iguais,~logo:\\\\\\5\!\!\!\backslash^{2y-2}~=~5\!\!\!\backslash^2\\\\\\2y-2~=~2\\\\\\2y~=~2+2\\\\\\2y~=~4\\\\\\y~=~\dfrac{4}{2}\\\\\\\boxed{y~=~2}$

c)

$125^{x}~=~5^6\\\\\\Utilizando~a~propriedade~da~potencia~de~potencia:\\\\\\\left(5^3\right)^x~=~5^6\\\\\\\boxed{5^{3x}~=~5^6}~~\Rightarrow~Igualdade~entre~potencias~de~mesma~base\\\\\\Pra~que~a~igualdade~seja~verificada,~os~expoentes~tambem~deverao\\ser~iguais,~logo:\\\\\\5\!\!\!\backslash^{3x}~=~5\!\!\!\backslash^6\\\\\\3x~=~6\\\\\\x~=~\dfrac{6}{3}\\\\\\\boxed{x~=~2}$

d)

$4^{z}~=~2\\\\\\Utilizando~a~propriedade~da~potencia~de~potencia:\\\\\\\left(2^2\right)^z~=~2^1\\\\\\\boxed{2^{2z}~=~2^1}~~\Rightarrow~Igualdade~entre~potencias~de~mesma~base\\\\\\Pra~que~a~igualdade~seja~verificada,~os~expoentes~tambem~deverao\\ser~iguais,~logo:\\\\\\2\!\!\!\backslash^{2z}~=~2\!\!\!\backslash^1\\\\\\2z~=~1\\\\\\\boxed{z~=~\dfrac{1}{2}}$